Introduction to supervised machine learning

Material covered in this lecture

• what is logistic regression?
• what cost function to use for logistic regression?
• how to train these models?

Logistic regression

• confusingly, it is a classifier not a regression (in the ML sense)
• it is used to create models to classify binary data
• very common across ML and statistics

Example data

Cancer and coins

Denote:

• \(y_i=0\) indicates cancer-free
• \(y_i=1\) indicates presence of lung cancer

Two outcomes: so is like flipping a coin!

How to model these data?

Coin flipping distribution

Denote:

• \(\text{Pr}(y_i=0) = 1-\theta\)
• \(\text{Pr}(y_i=1) = \theta\)

where \(0\leq \theta \leq 1\). We can represent in the following probability distribution:

\[\text{Pr}(y_i=y) = \theta^y(1-\theta)^{1-y} \]

which is known as the Bernoulli distribution:

\[\begin{equation} y_i \sim \text{Bernoulli}(\theta) \end{equation}\]

How to estimate coin bias?

Suppose you flip coin twice: \(y_1=1\), \(y_2=0\). Assuming independence:

\[\begin{equation} \text{Pr}(y_1=1,y_2=0|\theta) = \theta \times (1-\theta). \end{equation}\]

We call \(L(\theta)=\text{Pr}(y_1=1,y_2=0|\theta)\) a likelihood.

Maximum likelihood estimation

Want to choose \(\theta\) to maximise probability of obtaining those results

Derivatives

Find maximum by differentiating:

\[\begin{equation} \frac{d L}{d\theta} = 1 - 2 \theta = 0 \end{equation}\]

Rearranging, we obtain:

\[\begin{equation} \theta = \frac{1}{2} \end{equation}\]

How to model these data?

Biased coins

How to model bias?

In logistic regression, we use logistic function:

\[\begin{equation} \theta = \frac{1}{1 + \exp (-x)} \end{equation}\]

Logistic regression

We want to estimate how sensitive presence / absence of lung cancer is to tar, so model probability:

\[\begin{equation} \theta_i = f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_i))} \end{equation}\]

which is known as logistic regression and assume:

\[\begin{equation} y_i \sim \text{Bernoulli}(\theta_i) \end{equation}\]

How to estimate \(\beta_0\) and \(\beta_1\)?

Data for one individual \((x_i,y_i)\) has probability:

\[\text{Pr}(y_i=y) = f_\beta(x_i)^y(1-f_\beta(x_i))^{1-y} \]

Likelihood for two individuals’ data

Suppose we have data \((x_1,y_1=1)\) and \((x_2,y_2=0)\).

Assume data are:

• independent
• identically distributed

Then overall probability is just product of individual:

Likelihood for larger datasets

Same logic applies under i.i.d. assumption:

\[\begin{equation} L = \prod_{i=1}^{K} f_\beta(x_i)^{y_i} (1 - f_\beta(x_i))^{1 - y_i} \end{equation}\]

Maximum likelihood estimation

Unlike the simple coin flipping case, there is no analytic solution to the maximum likelihood estimates. Instead, do gradient descent:

\[\begin{align} \beta_0 &= \beta_0 - \eta \frac{\partial L}{\partial \beta_0}\\ \beta_1 &= \beta_1 - \eta \frac{\partial L}{\partial \beta_1} \end{align}\]

where \(\eta>0\) is the learning rate.

How to interpret model estimates?

Suppose we estimate that \(\beta_0=-1\) and \(\beta_1=2\). What do these mean?

\[\begin{equation} \theta_i = \frac{1}{1 + \exp (-(-1 + 2 x_i))} \end{equation}\]

so impact of incremental changes in \(x_i\) on the probability of lung cancer is nonlinear

Nonlinear impact

Can we find an interpretation?

\[\begin{equation} \theta_i = \frac{1}{1 + \exp (-(-1 + 2 x_i))} \end{equation}\]

meaning

\[\begin{equation} 1-\theta_i = \frac{\exp (-(-1 + 2 x_i))}{1 + \exp (-(-1 + 2 x_i))} \end{equation}\]

Calculate odds

The ratio of probability of lung cancer to probability of cancer-free is called odds:

\[\begin{align} \frac{\theta_i}{1-\theta_i} &=\exp (-1 + 2 x_i) \end{align}\]

so here \(\exp 2\approx 7.4\) gives the change to the odds for a one unit change in x_i. Because of this, \(\exp \beta_1\) is known as the odds ratio for that variable

Odds ratios

• if \(\beta_1 > 0\), the odds ratio \(>1\), which indicates that changes to a variable increase the probability of the \(y_i=1\) event occuring.
• vice versa for \(\beta_1 < 0\).

Log odds interpretation

Taking log of both sides:

\[\begin{equation} \log \frac{\theta_i}{1-\theta_i} = -1 + 2 x_i \end{equation}\]

so we see that \(\beta_1=2\) effectively gives the change to the log-odds for a one unit change in \(x_i\).

Multivariate logistic regression

straightforward to extend the model to incorporate multiple regressions:

\[\begin{equation} f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}))} \end{equation}\]

Logistic regression summary

• logistic regression models are binary classifiers (in ML speak)
• assumes Bernoulli distribution for outputs
• logistic function used to relate changes in inputs to outputs
• multivariate logistic regression is a commonly used tool